Preheat your oven to 350°F (175°C) and line baking sheets with parchment paper or silicone baking mats. Set aside for later.
In a small bowl, whisk together the flour, cornstarch, baking powder, baking soda, and salt until well combined. Set this dry mixture aside.
In a large mixing bowl using a hand mixer or stand mixer, beat the softened butter and granulated sugar on medium speed until well combined and slightly fluffy.
Add the egg and vanilla extract to the mixture and continue mixing until smooth and fully incorporated, scraping down the sides of the bowl as needed.
Reduce the mixer to low speed and gradually add the dry ingredients to the wet mixture. Mix just until a soft cookie dough forms—do not overmix.
Gently fold in the chopped chocolate sandwich cookies and white chocolate chips until they are evenly distributed throughout the dough.
Using a small cookie scoop (about 2 tablespoons of dough per cookie), portion out the dough and roll into balls with your hands. Lightly press each dough ball to slightly flatten the tops.
Place the cookies on the prepared baking sheets, spaced about 2 inches apart, and bake for 8–10 minutes, or until the edges are set. The centers may look slightly underbaked—that's perfect for a soft texture.
Allow the cookies to cool on the baking sheet for a few minutes before transferring them to a wire rack to cool completely. Enjoy your freshly baked cookies and cream cookies!